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Linear Regression Standard Error

Routine Summary Inputs Outputs Variables Used
Calculates the standard error associated with linear regression coefficients. L₁ - values of the independent variable
L₂ - values of the dependent variable		 Ans - a 2-element list containing the standard errors L₁, L₂, 
:2-Var Stats
:LinReg(ax+b)
:a√((r²ֿ¹−1)/(n-2)){1,√(Σx²/n)}

This routine computes the standard error (uncertainty) associated with the linear regression coefficients a and b (σa and σb, respectively) for the regression equation y=ax+b. Precisely stated, the true value of the coefficient a is expected to be within the interval a±σa, and similarly for b.

The routine returns a two-element list; σa is the first element, and σb is the second element.

If one prefers to use the function [LinReg(a+bx)](linreg(a-bx.html) instead of [LinReg(ax+b)](linreg(ax-b.html), the appropriate routine is:

:2-Var Stats
:LinReg(a+bx)
:b√((r²ֿ¹−1)/(n-2)){√(Σx²/n),1}

(note that the meanings of σa and σb have now interchanged).

In both routines, r², a, b, n, and Σx² are statistical variables.

Formulas

For the fitting equation y=ax+b,

\[\definecolor{darkgreen}{rgb}{0.90,0.91,0.859}\pagecolor{darkgreen} \begin{align*} \sigma_a&=a\sqrt{\frac{\frac1{r^2}-1}{n-2}} \\ \sigma_b&=\sigma_a\sqrt{\frac{\Sigma x^2}{n}} \end{align*}\]

where n is the number of data points, r² is the coefficient of determination, and Σx² is the sum of squares of the independent variable values.

Error Conditions

Reference

Lichten, William. Data and Error Analysis., 2nd. ed., Prentice Hall: Upper Saddle River, NJ, 1999.

Authors: KG